Name: how transfer functions shape circuit response
Uploaded: 2024-07-08T13:20:01.000Z
Duration: 1 min 25 s
Channel: JoVE

电路的频率响应表征了其在不同信号频率下的行为，可通过传递函数或网络函数进行分析。
该数学函数定义了输出与输入的比值。它可分为四种类型：电压增益、电流增益、转移阻抗和转移导纳。
在拉普拉斯域中，使用复变量's'分析电路的行为。
电路的传递函数通常可以用零点和极点来表示。
此处，零点是分子多项式的根，极点是分母多项式的根。在极点处，传递函数的值趋于无穷大；在零点处，传递函数的值为零。
考虑一个音频分频电路，该电路包含两个电容器和一个电感器，用于将来自放大器的高频信号选择性地传递给高音扬声器。
电感器、电阻器和电容器的组合提供了电路的阻抗。
根据欧姆定律，节点电压与输入电压成正比。
同样，输出电压与节点电压成正比。
最后，得到由输出电压与输入电压之比表示的传递函数。

电路的网络功能

Network Function of a Circuit

电路中的频率响应分析为了解电路在输入信号频率发生变化时的行为提供了重要见解。传递函数是一种数学工具，并且有助于理解这种行为。它定义了相位中输出和输入之间的关系，其中有四种类型：电压增益、电流增益、传递阻抗和传递导纳。传递函数中的关键组成部分是极点和零点。 
<img alt="Equation 1" src="/files/ftp_upload/15137/15137_Equation_1.svg" style="text-align: center; height: 32px;">
其中的零点是分子多项式的根，极点是分母多项式的根。
作为一个说明性示例，假设有一个音频分频电路能够将高音扬声器中的高频信号从放大器中分离出来。将“s”作为复变量，并在拉普拉斯域中对该电路的信号频率进行分析。
根据欧姆定律可以得知，输入电压与节点电压是相关的；随后，输出电压能够用输入电压来对其进行表示。这一分析过程能够得出传递函数，并且还表示了输出电压与输入电压的电压比。
<img alt="Equation 2" src="/files/ftp_upload/15137/15137_Equation_2.svg" style="text-align: center; height: 32px;">
频率响应分析具有广泛的应用，特别是在通信系统和控制系统中。用于无线电技术和电视技术等各种应用场景的电子滤波器通常会依靠这种分析来允许所需的信号通过，同时还会抑制不需要的信号。全面了解传递函数和频率响应对于电路设计和工程来说是至关重要的。
作为一种基本工具，传递函数简化了工程和电子学中对不同频率电路行为的评估。

电路的频率响应表征了其在不同信号频率下的行为，可通过传递函数或网络……进行分析

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Frequency Response

频率响应

KEY TERMS AND CONCEPTS

关键术语与概念

how transfer functions shape circuit response

Transfer functions show how a circuit responds as signal frequency changes. They describe the link between input and output in frequency response analysis. This makes them useful for studying how circuits behave in electronics and engineering.

A transfer function is a mathematical tool that relates phasor output to phasor input. It can be written as voltage gain, current gain, transfer impedance, or transfer admittance. The poles and zeros are the most important parts of the function. Zeros are the roots of the numerator polynomial, and poles are the roots of the denominator polynomial.

An audio crossover circuit gives a clear example of this idea. It separates high-frequency signals for a tweeter from an amplifier. To analyze this circuit, engineers work in the Laplace domain and use s as a complex variable. Ohm's Law is then used to connect the input voltage to the node voltage, and the output voltage is written in terms of the input voltage.

This process produces the transfer function as the output-to-input voltage ratio. Frequency response analysis is widely used in communication and control systems. Electric filters in radio and television also depend on it to let desired signals pass and reduce unwanted ones. A strong grasp of transfer functions helps with circuit design and with understanding how circuits act at different frequencies.

How Transfer Functions Shape Circuit Response

how-transfer-functions-shape-circuit-response

network function of a circuit

Transfer functions show how a circuit responds as signal frequency changes. They describe the link between input and output in frequency response analysis. This makes them useful for studying how circuits behave in electronics and engineering. A transfer function is a mathematical tool that relates phasor output to phasor input. It can be written as voltage gain, current gain, transfer impedance, or transfer admittance. The poles and zeros are the most important parts of the function. Zeros...

frequency response of a circuit

Inductive Circuit Frequency Response

Inductive circuit frequency response shows how a circuit with an inductor reacts to different frequencies. The analysis starts in the time domain and then moves into the frequency domain. In that step, the inductor is replaced by an impedance, and the circuit is written with phasors, which are sinusoidal signals shown with magnitude and phase. The transfer function describes how the output changes compared with the input at each frequency. A key parameter is the time constant, which is the...

gain

Circuit Gain and Phase Shift

Circuit gain and phase shift describe how a linear circuit changes a sinusoidal input voltage or current. These properties matter in circuits with reactive elements, because the response depends on the input frequency. As the frequency changes, the gain and phase shift also change. Gain compares the size of the output sinusoid to the size of the input sinusoid. If Vin is the input and Vout is the output, the gain K is the ratio of the output amplitude to the input amplitude. A gain greater...

bode plots

Reading Frequency Response on Bode Plots

Bode plots show how a circuit or system responds across frequency. They use a logarithmic frequency scale on the x-axis and gain in decibels on the y-axis. This format makes it easier to display a wide range of frequencies and to see how circuit parts affect behavior over that range. A network function gives the ratio of output to input for a system. Its complex form provides both magnitude and phase angle. The logarithmic gain on a Bode plot comes from the magnitude, calculated by multiplying...

transfer function and bode plots-i

Transfer Function Gain and Origin Effects

Transfer function gain and poles or zeros at the origin shape a system's Bode plot. The transfer function is often written in standard form after the polynomial coefficients are normalized. In that form, the constant gain, zeros, poles, and quadratic terms all appear clearly. The poles and zeros mark critical frequencies where the output magnitude and phase change noticeably. The constant gain K creates a fixed magnitude and phase term. Its magnitude is 20 log 10 K in decibels, and its phase...

transfer function and bode plots-ii

Bode Plot Slope and Phase Changes

Bode plots show how a transfer function changes with frequency. The standard form includes constant gain, poles and zeros at the origin, simple poles and zeros, and quadratic poles and zeros. Each part shapes the system response in a different way. A simple zero sets the magnitude trend first. The Bode magnitude plot stays flat at low frequencies, near 0 dB, and then rises at 20 dB per decade after the corner or break frequency, ω1. That corner frequency is where the straight-line...

bode plots construction

Reading Frequency Response with Bode Plots

Bode plots show the frequency response of a system with a magnitude plot and a phase plot. Both plots use a logarithmic frequency axis. In control system analysis, this makes it easier to see how a transfer function changes with frequency. To build a Bode plot, start with the transfer function H(ω). After normalization, the function includes constant gain, zeros, and poles. Each term affects the plot in a different way. A constant gain of 10 gives a starting magnitude of 20 dB, since 20...

series resonance

Series Resonance in RLC Circuits

Series resonance in an RLC circuit happens when the inductive reactance and capacitive reactance are equal. An RLC circuit contains a resistor, an inductor, and a capacitor. The impedance of the circuit is the ratio of the supply voltage to the circuit current. At resonance, the imaginary part of the impedance becomes zero. The frequency at which this occurs is called the resonant frequency. It depends on the inductance (L) and capacitance (C) of the circuit. Mathematically, the resonant...

characteristics of series resonant circuit

characteristics-of-series-resonant-circuit

Series Resonance and RLC Filter Behavior

Series resonance in an RLC circuit happens when the inductor, capacitor, and resistor are connected in series. At the resonant frequency, the inductive reactance and capacitive reactance are equal in size but opposite in sign, so they cancel each other. As a result, the circuit impedance becomes very small and is mainly set by the resistance. The resonant frequency of a series RLC circuit can be calculated from its inductance and capacitance. At this frequency, the current reaches its maximum...

parallel resonance

Parallel RLC Tuning at Resonance

A parallel RLC circuit uses a resistor, inductor, and capacitor connected to the same nodes. Because they share the same voltage, the circuit is analyzed with admittance, which describes how easily current can flow. In a parallel RLC circuit, admittance helps show how the circuit responds as the frequency changes. Resonance happens when the net reactance is zero. At that point, the capacitive and inductive effects cancel each other out. The resonant frequency is found from this condition. At...

frequency response of op amp circuits

Op Amp Gain and Bode Plot Basics

Operational amplifiers are used for signal conditioning, filtering, and mathematical tasks such as addition, subtraction, integration, and differentiation. Their frequency response shows how amplifier gain changes as the input frequency changes. This makes the gain of an op-amp a key idea in circuit design. At low frequencies, an op-amp can hold a constant gain called the dc gain, A0. As frequency rises, the gain starts to drop after the corner frequency, or break frequency, ω1. In other...

passive filters

Audio Frequency Control with Passive Filters

Passive filters shape the frequency spectrum of electrical signals. They use only passive parts such as resistors, inductors, and capacitors. Because they do not need an external power source, they can let selected frequency ranges pass or block others in many applications. Low-pass filters pass signals below the cutoff frequency, ωc, and reduce signals above it. In audio systems, they can send bass frequencies to woofers or subwoofers. For an RC low-pass filter, the transfer function H(s)...

active filters

Active Filter Design and Frequency Response

Active filters use op-amps, resistors, and capacitors to shape a signal's frequency response. They remove unwanted frequency components while keeping the parts of the signal that are needed. The design depends on the choice of resistor and capacitor values, which set the cutoff points and help define the overall response. A first-order low-pass active filter passes signals below a cutoff frequency and reduces signals above it. Its transfer function includes the low-frequency gain, also called...

scaling

Circuit Component Scaling for Filters

Circuit component scaling helps turn simple circuit values into practical ones for filter design and circuit analysis. Engineers often start with standard values like 1 ohm, 1 henry, or 1 farad because they make examples and problems easier to work through. Later, those values can be scaled to more realistic levels. This method is useful in filters, resonant circuits, and many other circuit analysis problems. It reduces calculation complexity while keeping the circuit behavior easy to study.

design example

Touch-Tone Filter Design Example

Touch-tone telephony uses dual-tone multi-frequency (DTMF) signaling and a filter design example to show how keypad tones are separated. It replaced the old rotary dial with a matrix-style keypad arranged in four rows and three columns. Each of the 12 buttons produces a unique signal pair. Every key press creates two sinusoidal tones at the same time. One tone comes from a low-frequency group between 697 and 941 Hz. The other comes from a high-frequency group between 1209 and 1477 Hz. This...

key terms and concepts

Learn how transfer functions describe circuit behavior across frequency, including poles, zeros, and an audio crossover example used in filters and electronics.

How Transfer Functions Shape Circuit Response | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

How Transfer Functions Shape Circuit Response

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